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Roberts’ example and the AR-property of convex sets in non-locally convex linear metric spaces. (English) Zbl 1003.54501
The paper begins with some still open, well-known problems in topology (the AR-problem, Schauder’s conjecture and the Banach problem of whether every infinite-dimensional compact convex set in a linear metric space is homeomorphic to the Hilbert cube). Then, in Section 2 the following characterization of ANR-spaces is given: A metric space is an ANR if and only if there is a zero sequence of open covers $$\{{\mathcal U}_n\}$$ of $$X$$ together with a map $$f:{\mathcal K}({\mathcal U})\to X$$ such that, for any map $$g:{\mathcal U}\to X$$ satisfying $$g(U)\in U$$, $$U\in{\mathcal U}$$, and for any sequence $$\{\sigma_k\}\subset{\mathcal K}({\mathcal U})$$ with $$n(\sigma_k)$$ converging to $$0$$, we have $$\text{diam}\{g(\sigma^0_k)\cup f(\sigma_k)\}\to 0$$. Here, $$\{{\mathcal U}_n\}$$ is a zero sequence if $$\text{mesh}({\mathcal U}_n)\to 0$$, $${\mathcal U}= \bigcup\{{\mathcal U}_n; n\in\mathbb{N}\}$$ and $${\mathcal K}({\mathcal U})= \bigcup\{N({\mathcal U}_n\cup{\mathcal U}_{n+1})$$; $$n\in\mathbb{N}\}$$, where $$N({\mathcal U}_n\cup{\mathcal U}_{n+1})$$ is the nerve of the cover $${\mathcal U}_n\cup{\mathcal U}_{n+1}$$ equipped with the Whitehead topology. Also, $$n(\sigma)= \sup\{n\in\mathbb{N}; \sigma\in N({\mathcal U}_n\cup{\mathcal U}_{n+1})\}$$ for each $$\sigma\in{\mathcal K}({\mathcal U})$$. In Section 3 some results and questions concerning Roberts’s examples of compact convex sets with no extreme points are reviewed. The last section is devoted to the following theorem: Let $$(X,\mu)$$ be a normalized measure space and let $$\phi$$ be an Orlicz function. Then $$L_\phi(\mu)= \{f: X\to\mathbb{R}$$, $$\int_X\phi(|f|)d\mu< \infty\}$$ with the $$F$$-norm $$\|f\|= \int_X\phi(|f|) d\mu$$ contains a needle point space if and only if $$\mu$$ is not purely atomic.
Reviewer’s comment: It is shown that the above-mentioned characterization of ANR’s (Theorem 2.1 in the paper under review) implies the Dugundji theorem that every convex set in a locally convex space is an AR. But it should be noted that in the authors’ proof of Theorem 2.1 some results from the monograph by S. Hu [Theory of retracts (1965; Zbl 0145.43003)] are used and the proof of these results (as given in Hu’s book) are based on the same technique developed by Dugundji for proving his famous theorem.
##### MSC:
 54E35 Metric spaces, metrizability 46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)